Q.6 If the radius of first Bohr orbit is 0.53 Å, then the radius of the third Bohr orbit is
(A) 2.12 Å (B) 4.77 Å (C) 1.59 Å (D) 3.18 Å

The radius of Bohr orbits scales with the square of the principal quantum number n.
For hydrogen, given the first orbit radius of 0.53 Å, the third orbit radius
is calculated using the formula:

r_n = r₁ × n²

This yields 4.77 Å as the correct answer for the third orbit.

Bohr Radius Formula

In the Bohr model, electron orbits have quantized radii given by:

r_n =

(n² h² ε₀) / (π m_e e²)

For hydrogen (Z = 1), this expression simplifies to:

r_n = r₁ × n²

where:

• r₁ = 0.53 Å (Bohr radius a₀)
• n = principal quantum number

For n = 3:

r₃ = 0.53 × 9 = 4.77 Å

Correct Answer Explanation

Substitute n = 3 into the formula:

r₃ = 0.53 × (3)² = 0.53 × 9 = 4.77 Å

This matches option (B) precisely, confirming the radius of the third Bohr orbit.

Why Other Options Are Incorrect

(A) 2.12 Å

This equals 0.53 × 4, which corresponds to n = 2 (since 2² = 4), not the third orbit.

(C) 1.59 Å

This equals 0.53 × 3, indicating linear scaling with n instead of n²,
which violates Bohr’s model.

(D) 3.18 Å

This equals 0.53 × 6, possibly arising from confusion with other atomic relations
such as velocity or energy, not orbital radius.

Option-wise Summary